Miloslav Feistauer

Space-Time Discontinuous Galerkin Method for the Problem of Linear Elasticity

The subject of this paper is the numerical solution of the problem of dynamic linear elasticity by several time-discretization techniques based on the application of the discontinuous Galerkin (DG) method in space. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of the elasticity term and the interior and boundary penalty are used. The DG space discretization is combined with the backward-Euler, second-order backward-difference formula and DG time discretization. Finally, we present some test problems.

Analysis of Space-Time DGFEM for the Solution of Nonstationary Nonlinear Convection-Diffusion Problems

The subject of this paper is the analysis of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear, convection-diffusion problems. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of diffusion terms and interior and boundary penalty are used. Then error estimates derived under a sufficient regularity of the exact solution are briefly characterized. The main attention is paid to the investigation of unconditional stability of the method. An important tool is the concept of the discrete characteristic function. The dominating convection case is not considered. Theoretical results are accompanied by numerical experiments.

The Interaction of Compressible Flow and an Elastic Structure Using Discontinuous Galerkin Method

In this paper we are concerned with the numerical simulation of the interaction of fluid flow and an elastic structure in a 2D domain. For each individual problem we employ the discretization by the discontinuous Galerkin finite element method (DGM). We describe the application of the DGM to the problem of compressible fluid flow in a time-dependent domain and also to the dynamic problem of the deformation of an elastic body. Finally, we present our approach to the coupling of these two independent problems: both are solved separately at a given time instant, but we require the approximate solutions to satisfy certain transient conditions. These transient conditions are met through several inner iterations. In each iteration a calculation of both the elastic body deformation problem and the problem of the compressible fluid flow is performed. The presented method can be applied to solve a selection of problems of biomechanics and aviation. Our numerical experiments are inspired by the simulation of airflow in human vocal folds, which implies the choice of the properties of the flowing fluid and the material properties of the elastic body. The results are post-processed in order to get a visualization of the approximate solution. We are especially interested in the visualization of the elastic body deformation and the visualization of some chosen physical quantities of the flow.

Numerical simulation of the interaction between a nonlinear elastic structure and compressible flow by the discontinuous Galerkin method

This paper is concerned with the numerical simulation of the interaction of compressible viscous flow with a nonlinear elastic structure. The flow is described by the compressible Navier-Stokes equations written in the arbitrary Lagrangian-Eulerian (ALE) form. For the elastic deformation the St. Venant-Kirchhoff model is used. In the space discretization the discontinuous Galerkin finite element method (DGM) is applied both for the flow problem in a time-dependent domain and for the dynamic nonlinear elasticity system. We show that the DGM is applicable to the discretization of both problems. As a new result we particularly present the application of the DGM to the discretization of the dynamic nonlinear elasticity problem and the DGM solution of the fluid-structure interaction (FSI). The applicability of the developed technique is demonstrated by several numerical experiments. The main novelty of the paper is the application of the DGM to the FSI problem using the model of compressible flow coupled with nonlinear elasto-dynamic system.